Sampling and Pulse Code Modulation Chapter 6

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1 Sampling and Pulse Code Modulation Chapter 6 Dr. Yun Q. Shi Dept of Electrical & Computer Engineering New Jersey Institute of Technology shi@njit.edu Sampling Theorem A Signal is said to be band-limited if g (t), its spectrum (FT) G(ω) G (ω) = 0 as ω > πb Sampling Theorem: The signal can be reconstructed from its samples taken uniformly at a rate R > B. That is, the minimum sampling frequency is fs = B [Ts = 1/B] (Ts: sampling interval) fs: Nyguist rate for g(t) Ts: Nyguist interval for g(t) Ts = 1/fs Dr. Shi Digital Communications 1

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4 Signal Reconstruction: Interpolation 1. Zero-order hold circuit (LPF) Figure 6.! " Simple interpolation using zero-order hold circuit (LPF) Frequency response of the LPF: ωt H( ω) = Ts sinc s 1 ω = sinc B 4B (6.8) Interpolation filter impulse response h(t): g ( t)* h( t) = g( t) Dr. Shi Digital Communications 7 Dr. Shi Digital Communications 8 4

5 Signal Reconstruction: Interpolation. Ideal interpolation (sinc function) Figure 6.3! "Ideal interpolation g ( t) = g ( kt s ) h( t kt s ) k = g ( kt s ) sin c[πb( t kt s )] Eq.6.10 k = g ( kt s ) sin c[πbt kπ )] k Dr. Shi Digital Communications 9 Dr. Shi Digital Communications 10 5

6 Maximum Information Rate: Statement Two pieces of information per second per Hertz bandwidth. That is, a maximum of B independent pieces of information per second can be transmitted error-freely, over a noise-less channel of bandwidth B Hz. Dr. Shi Digital Communications 11 Maximum Information Rate: Justification Noise free A channel of bandwidth B Hz can transmit a signal of bandwidth B Hz error-freely. Sampling Theorem: A signal of bandwidth B can be reconstructed from its Nyguist samples at a rate of B Hz. The signal can be reconstructed by B independent pieces of information per second (lower bound). Dr. Shi Digital Communications 1 6

7 Pulse Modulation PAM PWM PPM PCM Pulse Amplitude Modulation Pulse Width Modulation Pulse Position Modulation Pulse Code Modulation By far, the most popular among pulse modulation. Figure 6.8 shown next Dr. Shi Digital Communications 13 Dr. Shi Digital Communications 14 7

8 Dr. Shi Digital Communications 15 Time-Division Multiplexing (TDM) One advantage of using pulse modulation: It permits the simultaneous transmission of several signals on a time-sharing basis. Interweaving several signals along time domain. Figure 6.9 Dr. Shi Digital Communications 16 8

9 Dr. Shi Digital Communications 17 Quantization: Discretization of magnitude of a signal Input-output characteristic of a uniform midtread quantizer (Figure.4, Shi & Sun) Input-output characteristic of a uniform midrise quantizer (Figure.5, Shi & Sun, 1999) Y i = Q(x) if x belongs to (d i, d i+1 ) d i : decision levels i : index of intervals y i : reconstruction level (quantizing level) d = : step size di+ 1 i Dr. Shi Digital Communications 18 9

10 Quantization 1. Except for possibly the right-most and leftmost intervals, all intervals (hence, decision levels) along the x-axis are uniformly spaced (same interval length).. Except for possibly the outer intervals, the reconstruction levels are also uniformly spaced. Each inner reconstruction level is the arithmetic average of the two decision levels of the corresponding interval along x-axis. Dr. Shi Digital Communications 19 Dr. Shi Digital Communications 0 10

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12 Quantization Distortion Quant. noise, quant. error, e q = x Q(x) Mean-square quant. error MSE q = N i= 1 i+ 1 ( x Q( x)) N: number of intervals d d i f x (x): pdf x Q x) : e. ( ) ( q x e q f ( x) dx Dr. Shi Digital Communications 3 Quantization Distortion Assume that f x (x) is uniformly distributed MSE q = N = MSE 1 MSEq = σ q Variance of input x: d d i ( x Q( x)) q 1 dx N Dr. Shi Digital Communications 4 σ ( ) = N 1 x 1

13 Quantization Distortion Signal-to-noise ratio, SNR ms SNR σ log10 N x ms = log = σ q If n N = then n SNRms = log = 0 n log 6. 0ndb = Dr. Shi Digital Communications 5 Meaning: Quantization Distortion If we use the Natural Binary Code to code the reconstruction levels of a uniform quantizer with a uniformly distributed input source, then every increased bit in the coding brings out a 6.0 db increase in the SNR ms. That is, whenever the step size of the uniform quantizer decreases by a half, the MSE q decreases four times. Dr. Shi Digital Communications 6 13

14 Transmission Bandwidth Consider a binary PCM, (uniform quantization + NBC), There are N quantization levels, N = n. Using n binary digits (bits) to encode each quant. level. A signal m(t): band-limited to B Hz requiring a minimum of B samples per second. a total of n B bits/sec (bps) Dr. Shi Digital Communications 7 Transmission Bandwidth Because: A unit bandwidth (1 Hz) can transmit a maximum of two pieces of information per second, Need a minimum channel bandwidth B T Hz. B T = nb Hz Theoretical minimum transmission channel bandwidth required to transmit the PCM signal. Example 6. Dr. Shi Digital Communications 8 14

Time division multiplexing The block diagram for TDM is illustrated as shown in the figure

Time division multiplexing The block diagram for TDM is illustrated as shown in the figure CHAPTER 2 Syllabus: 1) Pulse amplitude modulation 2) TDM 3) Wave form coding techniques 4) PCM 5) Quantization noise and SNR 6) Robust quantization Pulse amplitude modulation In pulse amplitude modulation,